Question

How many permutations are there for the letters of the name Bathsheba? Solomon? Ahab? your own name?

Short answer

Bathsheba: 90720, Solomon: 2520, Ahab: 12, My name (example: 6 unique letters): 720

Step-by-step solution

Understanding Permutations

A permutation of a set is an arrangement of its elements in a sequence or order. The number of permutations of a set of elements is calculated differently if there are repeated elements versus when all elements are unique.

Counting Permutations for Bathsheba

The name 'Bathsheba' consists of 9 letters, where the letters B and A are repeated twice. The formula to find permutations when there are repeating elements is \( \frac{n!}{n_1! \times n_2! \times \ldots} \), where \(n\) is the total number of letters, and \(n_1, n_2, \ldots\) are the frequencies of the repeating letters. For Bathsheba, this is \( \frac{9!}{2! \times 2!} \).

Calculating Total Permutations for Bathsheba

Calculate \( 9! = 362880 \), and \( 2! \times 2! = 4 \). Thus, the number of permutations is \( \frac{362880}{4} = 90720 \).

Counting Permutations for Solomon

The name 'Solomon' consists of 7 letters, where the letter O appears twice. Using the same formula, we calculate the permutations as \( \frac{7!}{2!} \).

Calculating Total Permutations for Solomon

Calculate \( 7! = 5040 \), and \( 2! = 2 \). Thus, the number of permutations is \( \frac{5040}{2} = 2520 \).

Counting Permutations for Ahab

The name 'Ahab' consists of 4 letters, where the letter A appears twice. We use the formula \( \frac{4!}{2!} \).

Calculating Total Permutations for Ahab

Calculate \( 4! = 24 \), and \( 2! = 2 \). Thus, the number of permutations is \( \frac{24}{2} = 12 \).

Counting Permutations for My Own Name

Suppose your name consists of 6 unique letters (for example). The formula used is simply \( 6! \).

Calculating Total Permutations for My Own Name

Calculate \( 6! = 720 \). Therefore, there are 720 permutations of your name if all letters are unique.

Key concepts

Combinatorics

Combinatorics is a fascinating area of mathematics that deals with counting, arrangement, and combination of elements within sets. It helps us answer questions about how objects can be selected or organized. These concepts are prevalent in problems involving permutations and combinations. A permutation focus is on the different ways to arrange a set of elements in a sequence, where the order is crucial.

The fundamental idea behind combinatorics is understanding the number of possible ways to arrange elements in sets, especially when certain constraints or repetitions are involved. For example, calculating the permutations of a name involves considering each unique letter and any repeated elements.

You can often see combinatorics applied in day-to-day situations like scheduling tasks, organizing teams, and even in more complex fields such as cryptography and network theory.

Factorials

Factorials are a mathematical operation that is central to many combinatorial calculations. The factorial of a number, denoted by the symbol "!", is the product of all positive integers up to that number. For example, 4! equals 4 x 3 x 2 x 1, which is 24. Factorials are essential in permutations because they tell us the total number of ways to arrange a set of unique elements.

In permutation formulas, we often see the use of factorials when calculating different combinations of elements. For example, in determining the number of permutations for a name with repeated letters, such as "Bathsheba," you would use the factorial of the total count of letters and divide by the factorial of each set of identical letters.

By understanding factorials, you can easily compute a wide range of problems that require arranging or rearranging multiple elements.

Repeated Elements

When permutations involve repeated elements, special consideration is required. Repeated elements mean that some items within the set are identical, which reduces the total number of unique permutations because swapping identical items does not create a new arrangement. To account for repeated elements, we adjust the typical permutation formula.

The formula used in these cases is \( \frac{n!}{n_1! \times n_2! \times \ldots} \), where \( n \) is the total number of elements, and \( n_1, n_2, \ldots \) represent the factorials of the counts of each repeated element. For instance, in "Bathsheba," the right calculation considers the duplicate B and A letters through the division by their respective factorials.

This adjustment ensures that the count of permutations accurately reflects only the unique arrangements.

Discrete Mathematics

Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. This branch of math includes various topics such as graph theory, combinatorics, and number theory. Permutations fall under discrete mathematics because they deal with distinct and separate elements.

Within discrete mathematics, permutations are crucial as they represent different ways of arranging discrete items. Such studies help in understanding schedules, computer algorithms, and statistical probabilities. When dealing with names like "Solomon" or "Ahab," we delve into discrete structures because we're focused on finite and countable settings.

Learning permutations and their connection to discrete mathematics opens doors to more complex and diverse mathematical challenges and solutions.